The Inverse-Gap Law: A Proposal
Dynamic systems evolve at rates determined by their internal imbalances, revealing that stability and collapse are complementary expressions of a single recursive law of change. Cognition itself operates as a recursive dynamical process, where thought stability increases as conceptual imbalance decreases. Collapse—whether in quantum states, neural choices, or collective behavior—is not random but a lawful resolution of tension, its rate following an inverse relationship to the system’s energetic or informational disparity. Balance and adaptation are thus not opposites but reciprocal movements of the same feedback function, oscillating across the continuum of constraint and potential. Coherence emerges as a living equilibrium within this recursive interplay, suggesting that the universe, the mind, and intelligent systems alike evolve through the same fundamental principle of dynamic balance.
In short, The rate of change in any system is proportional to its deviation from balance.
This can be seen in the perpetual cycle of definable state changes:
Equilibrium→Perturbation→Amplification→Collapse→Re-equilibration→Emergence→Equilibrium...Etc.
This loop can be described mathematically as a recursive feedback attractor, or narratively as the cycle of balance and adaptation.
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% ---------- Title ----------
\title{\textbf{Recursive Generative Emergence:\\
A Universal Collapse--Emergence Proposition Bridging Physics, Cognition, and Coherence}\\[0.5em]
\large Interpreting Penrose's Objective Reduction as a Limiting Case within a Recursive Dynamical Framework}
\author{C.\,L.\,Vaillant\\
\textit{Independent Researcher, rgemergence.com}\\
\texttt{codyvaillant@gmai.com}}
\date{October-9-2025}
\begin{document}
\maketitle
\noindent\textbf{Keywords:} recursive dynamics, objective reduction, quantum gravity, cognitive coherence, complex systems, artificial intelligence, falsifiability.\\
\textbf{arXiv Categories:} quant-ph, cs.AI, physics.gen-ph, nlin.AO
% ---------- Abstract ----------
\begin{abstract}
Recursive Generative Emergence (RGE) proposes a general dynamical proposition linking energy or information imbalance to coherence stability across physical, cognitive, and computational systems. The central law expresses coherence change as a bilinear differential relation, $dC/dt=k(P-P_c)(\Psi-\Psi_c)$, where $C$ is an action-like coherence measure, $P$ a structural field, and $\Psi$ a potential field. Imposing an action-like threshold $\hbar_{\mathrm{eff}}$ yields a universal collapse or convergence time $\tau=\hbar_{\mathrm{eff}}/(k\Delta\Phi)$, with $\Delta\Phi=(P-P_c)(\Psi-\Psi_c)$. When $\Delta\Phi$ represents gravitational self-energy, this relation approximates Penrose's Objective Reduction (OR) as a limiting correspondence under similar energy-balance assumptions. We provide operational definitions across domains, a dimensionally correct worked example, a comparison with GRW and CSL models, concrete empirical protocols for physics, cognition, and AI, and a minimal Python prototype. RGE is presented as a testable proposition with explicit falsification criteria, an analogical ethical framing, and scope limits that avoid overclaiming.
\end{abstract}
% ---------- Significance ----------
\section*{Significance Statement}
This work frames gravitational objective reduction as one limit of a broader recursive feedback law for coherence loss. It provides measurable mappings, testable predictions, and a prototype computational harness, enabling cross-domain checks of a single inverse-gap scaling relation.
% ---------- Introduction ----------
\section{Introduction}
Penrose's Objective Reduction (OR) \cite{Penrose1994,Penrose2004} links quantum state collapse to gravitational self-energy $E_G$, predicting $\tau=\hbar/E_G$. Stochastic collapse models such as GRW \cite{GRW1986} and CSL \cite{Pearle1989} introduce noise-driven localization, while Diósi's approach \cite{Diosi1989} adds gravitationally motivated stochasticity. In cognitive science and machine learning, stability is often modeled as predictive balance (e.g., the free-energy principle \cite{Friston2010}). This paper proposes \emph{Recursive Generative Emergence} (RGE), a single bilinear feedback law governing coherence change across these domains, with explicit units, operational definitions, and falsifiable empirical predictions.
% ---------- Mathematical Foundations ----------
\section{Mathematical Foundations}
\subsection{Recursive Law and Definitions}
We posit the bilinear rate law
\begin{equation}
\label{eq:rge}
\frac{dC}{dt}=k\,(P-P_c)(\Psi-\Psi_c)\equiv k\,\Delta\Phi,
\end{equation}
where $C(t)$ is an action-like coherence measure (units J$\cdot$s), $P$ is a structural field (constraint/rigidity), $\Psi$ is a potential field (capacity/openness), $k$ is a coupling constant, and $\Delta\Phi$ is a generalized energy-like gap. Collapse or convergence occurs when the integrated action reaches a threshold $\hbar_{\mathrm{eff}}$:
\begin{equation}
\label{eq:threshold}
\int_{t_0}^{t_0+\tau} k\,\Delta\Phi\,dt=\hbar_{\mathrm{eff}}
\quad\Rightarrow\quad
\tau \approx \frac{\hbar_{\mathrm{eff}}}{\langle k\,\Delta\Phi \rangle}
\end{equation}
where $\langle\cdot\rangle$ denotes the time-average over $[t_0,t_0+\tau]$. For quasi-stationary $\Delta\Phi$, $\langle k\Delta\Phi\rangle\approx k\Delta\Phi$.
\subsection{Dimensional Consistency}
Assume $[P]=\mathrm{J/m^3}$ and $[\Psi]=\mathrm{m^3}$ so that $[(P-P_c)(\Psi-\Psi_c)]=\mathrm{J}$. Then $[k]=1$ (dimensionless), $[dC/dt]=\mathrm{J}$, and $[C]=\mathrm{J\cdot s}$, preserving the units of action. In non-physical domains where $P$ or $\Psi$ take informational forms, $k$ inherits compensating units to maintain $[dC/dt]=\mathrm{J}$ and $[C]=\mathrm{J\cdot s}$.
\subsection{Relation to Penrose OR}
Identifying $\Delta\Phi$ with gravitational self-energy $E_G$ and taking $k=1$\footnote{For self-gravitating configurations, the gravitational coupling saturates dimensional balance, making $k=1$ a natural normalization. In other domains, an effective $k_{\mathrm{eff}}$ applies.} recovers the same dimensional form as Penrose's $\tau=\hbar/E_G$. This is a \emph{limiting correspondence under similar energy-balance assumptions}, not a derivation.
\subsection{Entropy Flow and Feedback Polarity}
Define $S=-\partial C/\partial\Psi$. Differentiating \eqref{eq:rge} with respect to $\Psi$ shows
\begin{equation}
\frac{dS}{dt}=-\frac{\partial}{\partial\Psi}\left[k(P-P_c)(\Psi-\Psi_c)\right],
\end{equation}
so aligned deviations ($P>P_c$, $\Psi>\Psi_c$) increase entropy production (positive feedback), while opposing deviations reduce it (negative feedback), linking energy exchange to information flow.
% ---------- Operational Definitions ----------
\section{Operational Definitions and Scope}
\begin{table}[h!]
\centering
\caption{Operational mapping of RGE variables across domains.}
\adjustbox{max width=\textwidth}{
\begin{tabular}{@{}llll@{}}
\toprule
Domain & $P$ (structural) & $\Psi$ (potential) & Gap $\Delta\Phi$ \\
\midrule
Quantum physics & Mass density / curvature & Gravitational potential & $E_G=\int \rho(\mathbf r)\,\Delta g\,d^3r$ \\
Cognition & Neural stability / synaptic gain & Prediction error / expectation & $\Delta I$ (change in Shannon info) \\
Machine learning & Weight constraint / regularization & Loss curvature / potential & $\Delta\Phi$ (cross-entropy or KL decrease) \\
Complex systems & Structural rigidity / constraints & Adaptive flow capacity & Resource imbalance \\
\bottomrule
\end{tabular}
}
\end{table}
\subsection{Scope and Limitations}
RGE proposes a \emph{form} for recursive coherence dynamics; it does not derive quantum mechanics or consciousness. Parameters $k$ and $\hbar_{\mathrm{eff}}$ are domain-specific and empirically determined. Universality refers to structure, not identical constants across systems.
% ---------- Analytical Examples and Numerical Demonstrations ----------
\section{Analytical Examples and Numerical Demonstrations}
\subsection{Harmonic Oscillator (Dimensional Correction)}
For a mass $m$ and angular frequency $\omega$, the energy at amplitude $x$ is $E=\tfrac{1}{2}m\omega^2 x^2$. Approximating $\langle \Delta\Phi\rangle\simeq E$ over $\tau$ and imposing $E\,\tau=\hbar_{\mathrm{eff}}$ with \eqref{eq:threshold} yields
\begin{equation}
x_c=\sqrt{\frac{2\hbar_{\mathrm{eff}}}{m\omega\,k}}.
\end{equation}
Example: $m=10^{-15}\,\mathrm{kg}$, $\omega=10^{6}\,\mathrm{s}^{-1}$, $\hbar_{\mathrm{eff}}=\hbar$, $k=1$ gives $x_c\simeq 1.4\times 10^{-12}\,\mathrm{m}$, a realistic mesoscopic scale.
\subsection{Cognitive Analogue}
For a two-alternative decision with information difference $\Delta I$, the predicted reaction time scales as
\begin{equation}
T_r \simeq \frac{\hbar_{\mathrm{eff}}}{k\,\Delta I},
\end{equation}
implying faster decisions under larger informational gaps.
\subsection{AI Analogue}
For a neural network minimizing cross-entropy $L$, define a per-epoch gap $\Delta\Phi = L_{n-1}-L_n$. The RGE scaling predicts
\begin{equation}
\tau \propto \frac{1}{\Delta\Phi},
\end{equation}
where $\tau$ is epochs to reach a criterion (e.g., 95\% validation accuracy). A linear fit of $\tau$ versus $\Delta\Phi^{-1}$ estimates $(k/\hbar_{\mathrm{eff}})^{-1}$.
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=1.0]
\draw[->] (0,0) -- (7,0) node[right] {$t$};
\draw[->] (0,0) -- (0,3) node[above] {$C(t)$};
\draw[domain=0:6,smooth,variable=\x,blue,thick] plot ({\x},{2*(1-exp(-0.7*\x))*exp(-0.2*\x)});
\draw[dashed,red] (0,2.5) -- (6.2,2.5) node[right,black] {$|\Delta C|=\hbar_{\mathrm{eff}}$};
\node at (3,2.8) {\small illustrative threshold};
\end{tikzpicture}
\caption{Illustrative conceptual trajectory of $C(t)$ approaching $\hbar_{\mathrm{eff}}$ (schematic, not fitted data).}
\end{figure}
% ---------- Model Comparison ----------
\section{Comparison with Existing Collapse Models}
\begin{table}[h!]
\centering
\caption{Key distinctions among leading collapse models.}
\adjustbox{max width=\textwidth}{
\begin{tabular}{@{}lllll@{}}
\toprule
Model & Collapse Time & Physical Basis & Domain & Testable Signature \\
\midrule
Penrose OR & $\tau=\hbar/E_G$ & Gravity (self-energy) & Physical & Mass-dependent coherence loss \\
GRW & $\lambda^{-1}$ & Stochastic localization & Universal & Constant rate, mass-independent \\
CSL & $(\lambda N)^{-1}$ & Continuous localization & Physical & Mass and particle-number scaling \\
RGE & $\hbar_{\mathrm{eff}}/(k\Delta\Phi)$ & Recursive feedback & Cross-domain & Inverse-gap scaling across $\Delta\Phi$ \\
\bottomrule
\end{tabular}
}
\end{table}
% ---------- Integration with Recursive and Symbolic Layers ----------
\section{Integration with Recursive and Symbolic Layers}
\paragraph{RCM (Recursive Collapse Model).} RCM models hierarchical sequences:
\[
\text{Emergence} \rightarrow \text{Coherence} \rightarrow \text{Instability} \rightarrow \text{Collapse} \rightarrow \text{Re-emergence}.
\]
Each micro-collapse obeys \eqref{eq:rge}; ensemble statistics determine macro-stability.
\paragraph{SAC (Symbolic Abstraction Core).} Symbolic stabilization follows
\[
\dot{C}_{\text{symbol}}=k_s(S-S_c)(M-M_c),
\]
where $S$ and $M$ denote syntactic and semantic coherence. Interpretation is a symbolic collapse event.
% ---------- Ethical and Philosophical Context ----------
\section{Ethical and Philosophical Context (Analogical Extension)}
The Justice--Cooperation--Balance (J--C--B) triad is an \emph{analogy}, not a mathematical consequence of RGE: Justice as proportional fairness (feedback conservation), Cooperation as reciprocal coupling, and Balance as dynamic equilibrium. No metaphysical claims are made; the analogy motivates fair, stability-preserving design in socio-technical systems.
% ---------- Empirical and Experimental Pathways ----------
\section{Empirical and Experimental Pathways}
\subsection{Testable Predictions}
\begin{enumerate}
\item Inverse-gap scaling: $\tau \propto 1/\Delta\Phi$ across domains.
\item Threshold quantization: coherence loss when $\int k\,\Delta\Phi\,dt=\hbar_{\mathrm{eff}}$.
\item Reciprocity stability: maximal at $P\approx P_c$ and $\Psi\approx \Psi_c$.
\end{enumerate}
\subsection{Physics}
Levitated nanospheres and optomechanical interferometers can test $\tau\propto 1/E_G$. Deviations from pure stochastic predictions at low mass would support recursive coupling.
\subsection{Cognition}
Two-alternative forced-choice with EEG/MEG: estimate $\Delta I$ (predictive information difference) between alternatives; test $T_r \propto 1/\Delta I$. Fit slope $\approx \hbar_{\mathrm{eff}}/k$.
\subsection{Artificial Intelligence}
Train CNNs on MNIST or CIFAR-10. Define $\tau$ as epochs to reach 95\% validation accuracy. Compute $\Delta\Phi$ as mean cross-entropy decrease per epoch. Test linearity of $\tau$ versus $1/\Delta\Phi$; recover $k$ from slope.
\subsection{Complex Systems}
Agent-based simulations with $P$ as structural rigidity and $\Psi$ as adaptive flow predict that collapse frequency scales with resource gap $\Delta\Phi$.
\subsection{Falsification}
RGE is falsified if (i) collapse times do not show inverse-gap scaling, (ii) reciprocity does not correlate with stability, or (iii) simpler stochastic models fit data better.
% ---------- Appendices ----------
\section{Appendices}
\subsection*{Appendix A: Symbol Glossary}
\begin{table}[h!]
\centering
\caption{Principal symbols and units.}
\adjustbox{max width=\textwidth}{
\begin{tabular}{@{}lll@{}}
\toprule
Symbol & Definition & Units \\
\midrule
$C(t)$ & Coherence (action-like) & J$\cdot$s \\
$P,\Psi$ & Structural / potential variables & J/m$^3$, m$^3$ \\
$k$ & Coupling constant & Context-specific \\
$\Delta\Phi$ & Energy/information gap & J \\
$\hbar_{\mathrm{eff}}$ & Domain action threshold & J$\cdot$s \\
$\tau$ & Collapse/convergence time & s \\
$\Delta I$ & Information gap & bits or nats \\
\bottomrule
\end{tabular}
}
\end{table}
\subsection*{Appendix B: Recursive Cascade Summary}
\[
\text{RGE (energetic)} \Rightarrow
\text{RCM (temporal)} \Rightarrow
\text{SAC (symbolic)} \Rightarrow
\text{J--C--B (ethical)}.
\]
\subsection*{Appendix C: Python Prototype (AI Test)}
\begin{lstlisting}[language=Python, basicstyle=\ttfamily\small]
import numpy as np
import matplotlib.pyplot as plt
# Synthetic RGE-style learning simulation
epochs = np.arange(1, 101)
delta_phi = np.linspace(0.1, 1.0, 100) # effective loss gap per epoch
hbar_eff, k = 1.0, 1.0
tau = hbar_eff / (k * delta_phi) # predicted inverse-gap scaling
# Add mild noise for realism
rng = np.random.default_rng(42)
tau_noisy = tau + rng.normal(0, 0.05 * tau, size=tau.shape)
plt.figure(figsize=(6,4))
x = 1.0 / delta_phi
plt.scatter(x, tau_noisy, s=12, label='simulation')
fit = np.polyfit(x, tau_noisy, 1)
plt.plot(x, np.polyval(fit, x), '--', label=f'linear fit (slope={fit[0]:.2f})')
plt.xlabel(r'$1/\Delta\Phi$')
plt.ylabel(r'$\tau$ (epochs)')
plt.title('RGE inverse-gap scaling test (synthetic)')
plt.legend()
plt.tight_layout()
plt.show()
\end{lstlisting}
This toy simulation illustrates the expected linear $\tau$--$\Delta\Phi^{-1}$ relationship under RGE, serving as a minimal reproducible test harness.
% ---------- References ----------
\begin{thebibliography}{9}
\bibitem{Penrose1994} R. Penrose, \emph{Shadows of the Mind}, Oxford University Press, 1994.
\bibitem{Penrose2004} R. Penrose, \emph{The Road to Reality}, Jonathan Cape, 2004.
\bibitem{GRW1986} G.\,C. Ghirardi, A. Rimini, and T. Weber, ``Unified dynamics for microscopic and macroscopic systems,'' \emph{Phys. Rev. D}, 34, 470--491 (1986).
\bibitem{Pearle1989} P. Pearle, ``Combining stochastic dynamical state-vector reduction with spontaneous localization,'' \emph{Phys. Rev. A}, 39, 2277--2289 (1989).
\bibitem{Diosi1989} L. Diósi, ``Models for universal reduction of macroscopic quantum fluctuations,'' \emph{Phys. Rev. A}, 40, 1165--1174 (1989).
\bibitem{Bassi2013} A. Bassi, K. Lochan, S. Satin, T.\,P. Singh, and H. Ulbricht, ``Models of wave-function collapse, underlying theories, and experimental tests,'' \emph{Rev. Mod. Phys.}, 85, 471--527 (2013).
\bibitem{Friston2010} K. Friston, ``The free-energy principle: a unified brain theory?,'' \emph{Nat. Rev. Neurosci.}, 11, 127--138 (2010).
\end{thebibliography}
\end{document}
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% Internal audit (for generator):
% - Dimensional analysis: checked
% - Claim calibration: proposition/limiting correspondence; no derivation claimed
% - Comparative table: included (Penrose, GRW, CSL, RGE)
% - Empirical hooks: physics, cognition, AI; falsification criteria specified
% - Operational definitions: table included
% - Width safety: adjustbox wraps all tables; tabcolsep/arraystretch tuned
% - Figure: TikZ schematic compiles within textwidth
% - References: all cited; thebibliography used; ASCII-safe
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